Note that the inequality constraint in (REF ) is convex [1]}.
For squared error loss, the objective function \(\mathcal {E}(g^{(\mathbf {w})})\)  is also convex.
Thus, for linear least squares regression, we can reformulate (REF )
as a dual unconstrained problem [1]}
\(\hat{g} \in \operatornamewithlimits{arg\;min}_{\mathbf {w} \in \mathbb {R}^{n},\alpha \in \mathbb {R}} \mathcal {E}(g^{(\mathbf {w})}) + \lambda {\rm E}\big \lbrace  \big (\mathbf {w}^{T}\mathbf {x}- \alpha \hat{u} \big )^{2}\big \rbrace .\) 
